Problem: Simplify to lowest terms. $\dfrac{30}{40}$
Solution: There are several ways to tackle this problem. What is the greatest common factor (GCD) of 30 and 40? $30 = 2\cdot3\cdot5$ $40 = 2\cdot2\cdot2\cdot5$ $\mbox{GCD}(30, 40) = 2\cdot5 = 10$ $\dfrac{30}{40} = \dfrac{3 \cdot 10}{ 4\cdot 10}$ $\hphantom{\dfrac{30}{40}} = \dfrac{3}{4} \cdot \dfrac{10}{10}$ $\hphantom{\dfrac{30}{40}} = \dfrac{3}{4} \cdot 1$ $\hphantom{\dfrac{30}{40}} = \dfrac{3}{4}$ You can also solve this problem by repeatedly breaking the numerator and denominator into common factors. For example: $\dfrac{30}{40}= \dfrac{2\cdot15}{2\cdot20}= \dfrac{2\cdot 5\cdot3}{2\cdot 5\cdot4}= \dfrac{3}{4}$